Linear Response

ε_λ2σ;l(t)hψ_λ^†

1l+pψ_σlb_pit−ε˜_σλ1;l+p(t)hψ^†_σl+pψ_λ2lb_pit

o

+n g_p^λ2hψ_λ^†

1l+pψ_λ2l+pit−g_p^λ1hψ^†_λ

1lψ_λ2lit

ohb^†_pb_pit

+g^λ_p²hψ_λ^†

1l+pψ_λ2l+pit−X

σ

g_p^σhψ^†_λ

1l+pψ_σl+pithψ_σl^†ψ_λ2lit

+

δλ2;l−δ^∗_λ1;l+p hψ_λ^†_1l+pψλ2lbpi^t

can be obtained. The time behavior of the phonon distribution function which appears on the right-hand side of Equation2.81is determined by the same differential equation as in Section2.2.

The system of kinetic equations which has been derived in this section corresponds to a system of kinetic equations which has been derived by means of the equations-of-motion method, if the electron-electron interaction is treated within the framework of the Hartree-Fock approximation [36]. In comparison with the results from Sec-tion 2.2 two differences can be observed. First, the products of first order phonon-assisted densities and ordinary electronic densities which appear in the collision term δhψ_λ^†

1l+pψ_λ2lb_piee in Equation 2.24 do not appear on the right-hand side of Equation 2.81. In the language of Feynman diagrams these terms can be identified as corrections to the electron-phonon interaction vertex which are due to multiple electron-electron scattering processes [54]. In addition to that the two phonon coherence hb_−pb_pit is missing. However, the absolute value of this function is usually small in compari-son with the phonon distribution function. Therefore the omission of the two phonon coherence should not alter the dynamics of the system significantly. In this context it should also be mentioned that the time behavior of the phonon functions is often neglected completely by replacing the time dependent functions with their initial val-ues [36, 43] without changing the results of the numerical calculations substantially.

The consequences of the neglect of the vertex corrections, however, have not been the object of a closer investigation, yet. In the following, the dynamics of the system will therefore be calculated by using the kinetic equations from Section2.2with and without vertex corrections in order to find out if significant differences exist between them.

2.5 Linear Response

It turns out to be useful to linearize the kinetic equations from Section2.2with respect toE(t)in order to achieve a better understanding of the influence of the vertex correc-tions. Within this approximation only the off-diagonal elements of the density matrix

are dynamical quantities. The time behavior of the polarizationhψ_vl^†ψ_clitis determined

The first order phonon-assisted densities which appear as source terms on the right-hand side of Equation2.82satisfy the equations

id

where the correction terms have been omitted since they would not affect the results of the following investigations. The optical states with no center-of-mass momen-tum hψ_vl^†ψcli^t are coupled to the two subspaces which are formed by the so-called dark states hψ^†_vl+pψ_clb_pit and hψ_vl−p^† ψ_clb^†_pit by phonon emission and absorption pro-cesses respectively. While the Hartree-Fock contributions lead to a constant shift of the free one-particle energies the vertex corrections are responsible for the last terms on the right-hand sides of Equation 2.83 and Equation 2.84 which are necessary for the correct description of the excitonic scattering dynamics in the two dark subspaces.

Without them the pairs of conduction electrons and valence holes with a non-vanishing

center-of-mass momentum would be treated as non-interacting particles unlike the op-tically excited electron-hole pairs.

The kinetic equations for the first order phonon-assisted densities are equivalent to the following integral equations

hψ_vl+p^† ψ_clb_pit= (−i) g

√N n

1 +n_B(ω_LO)o

(2.85)

×X

k

Z t t0

dτn

G⁺_l;k−p(p;t−τ)−G⁺_l;k(p;t−τ)o

hψ_vk^† ψ_ckiτ

and

hψ_vl^†_−pψ_clb^†_pit= (−i) g

√Nn_B(ω_LO) (2.86)

×X

k

Z t t0

dτn

G⁻_l;k+p(p;t−τ)−G⁻_l;k(p;t−τ)o

hψ^†_vkψ_ckiτ. The Green’s functions in Equations2.85and2.86describe the dynamics of the first or-der phonon-assisted densities in the absence of the electron-phonon interaction. They obey the following differential equation

id dtG^±_l

1;l2(p;t) = X

k

Ω^±_l

1;k(p)G^±_k;l

2(p;t) (2.87)

with the initial condition

G^±_l

1;l2(p; 0) =δ_l1;l2. (2.88) If the long-range part of the electron-electron interaction in Equation2.11 vanishes, the matrix elements read

Ω^±_l

1;l2(p) =δ_l1;l2{E(l₁,±p)±ω_LO} − U

N. (2.89)

The functionE(l, p)is defined by the relation

E(l, p) =_c;l−_v;l+p+U (2.90)

=E_G(U)−T(p) cos(l+ϕ(p))

and describes the energy of an excited electron-hole pair whose total momentum is equal to−pwhere the band gap is shifted because of the electron-electron interaction.

The generalized hopping elementT(p) and the phase shift function ϕ(p) satisfy the equations

T(p) = 2p

t²_v +t²_c + 2t_ct_vcos(p) (2.91)

and

ϕ(p) = arctan

t_vsin(p) t_c+t_vcos(p)

(2.92) while the functionE_G(U), which determines the position ofE(l, p)in the energy spec-trum, is given by

E_G(U) = 2t_c + 2t_v + ∆ +U. (2.93) In order to solve the differential equation for G^±_l1;l2(p;t), it is sufficient to solve the corresponding eigenvalue problem for a conduction electron and a valence hole in the presence of an attractive interaction. If the interaction function has only an on-site component, the eigenenergies and eigenstates can be calculated analytically and the explicit form of the Green’s functions is known:

G⁺_l

1;l2(p;t) = e^−iωLOtG_l1;l2(p;t) = e^−iωLOthl₁, p|e^−itH(p)|l₂, pi, (2.94) G⁻_l

1;l2(p;t) = e^+iωLOtG_l1;l2(−p;t) = e^+iωLOthl₁,−p|e^−itH(p)|l₂,−pi. (2.95) The definitions for the expressions which appear on the right-hand sides of Equations 2.94 and2.95 can be found in Appendix Btogether with the complete analytical so-lution of the eigenvalue problem. If the long-range interaction parameter U˜ did not vanish, the differential equation for the Green’s functions could only be solved numer-ically.

If the first order phonon-assisted densities in the differential equation for the in-terband polarization are replaced by the expressions which appear on the right-hand sides of Equations 2.85 and 2.86, the following integro-differential equation for the components of the interband polarization

id

dthψ_vl^†ψ_clit=E(l,0)hψ_vl^†ψ_clit− U N

X

k

hψ_vk^† ψ_ckit−dE(t) (2.96)

+X

k

Z _∞

−∞

dτS¯_l;k(t−τ)hψ^†_vkψ_ckiτ

is obtained in the limit t₀ → −∞ where the memory function S¯_l1;l2(t) satisfies the relation

S¯_l1;l2(t) = (−i)g² N

{1 +n_B(ω_LO)}e^−iωLOt+n_B(ω_LO)e^iωLOt θ(t) (2.97)

×X

q6=0

{G_l1;l2(q;t) +G_{l1−q;l2−q}(q;t)−G_l1−q;l2(q;t)−G_l1;l2−q(q;t)}. The first three terms on the right-hand side of Equation 2.96 describe the undamped dynamics of an optically excited electron-hole pair with a vanishing center-of-mass

momentum while the last term is responsible for the decay of the excited state due to phonon emission and absorption processes.

The expression in Equation 2.96 could be used as a starting point for a detailed numerical calculation of the linear response of the semiconductor. However, since the interest is mainly focused on the effect of the vertex corrections, the integro-differential equation for the interband polarization is now solved analytically by neglecting all contributions to the polarization which come from the continuum states. Then the dynamics of the components of the interband polarization only depends on the time behavior of the excitonic state

hψ_vl^†ψ_clit = Φ^ex_l P_ex(t) (2.98) where the definition for the vector components of the excitonic eigenstate can be found in AppendixB. By means of Equation 2.96 the following identity for the excitonic polarization

P_ex(t) =dX

k

Φ^ex_k Z t

−∞

dτχ¯_ex(t−τ)E(τ) (2.99)

can be derived. The excitonic susceptibilityχ¯_ex(t)satisfies the differential equation id

dtχ¯_ex(t) = ω_ex(0) ¯χ_ex(t) + Z _∞

0

dτS¯_ex(t−τ) ¯χ_ex(τ) (2.100) with the initial condition χ¯_ex(0) = −1. The explicit expression for the energy of the exciton with no center-of-mass momentum ω_ex(0) can be found in Appendix B.

The functionS¯_ex(t)is the diagonal element of the matrixS¯_l1;l2(t)with respect to the excitonic state. It is given by

S¯_ex(t) = (−i)g² N

{1 +n_B(ω_LO)}e^−iωLOt+n_B(ω_LO)e^iωLOt θ(t) (2.101)

×X

q6=0

X

k1k2

Φ^ex_k

1+q−Φ^ex_k

1

Φ^ex_k

2+q−Φ^ex_k

2 G_k1;k2(q;t).

The integro-differential equation for the susceptibilityχ¯_ex(t)can be solved easily with the help of a Fourier transformation:

χ_ex(ω) = Z _∞

0

dτ e^iωtχ¯_ex(t) =− 1

ω+i0−ω_ex(0)−S_ex(ω+i0). (2.102) If the electrons and phonons did not interact with each other, the denominator in Equa-tion2.102 would simply describe a resonance at the exciton energy ωex(0). The in-fluence of the electron-phonon scattering processes is taken into account by the

self-energy functionS_ex(ω+i0)which is defined by the relation This self-energy function can be split up into the contributions of the phonon emission and the phonon absorption processes. Both contributions have the same form, only their prefactors and their positions in the spectrum with respect toω_ex(0)are different.

As the spectrum of the dark states hψ^†_vl+pψ_clb_pit is located above the exciton energy their influence on the form ofχ_ex(ω)is negligible in comparison with the influence of the dark stateshψ^†_vl−pψ_clb^†_pit.

In Figures2.5and2.6the functionS_ex(ω)has been plotted using the kinetic equa-tions with and without vertex correcequa-tions for a system with N = 2000 elementary cells. The model parameters are those which have already been listed in Section2.1 and the energyωis measured with respect to the renormalized band gap∆ = ∆ +˜ U.

The self-energy is only depicted in the vicinity of the energyω_ex(0)≈ −0.4ω_LO since the form of χ_ex(ω) is mainly determined by the behavior of the denominator from Equation2.102near the exciton energy. If the self-energy is calculated with the vertex corrections, its imaginary part exhibits a sharp peak at the energyω≈ −0.9ω_LO. This peak denotes the position of the upper edge of the excitonic band for the dark states which are coupled to the optically excited exciton by phonon absorption processes.

For energies which are smaller than the energy of the lower edge of this excitonic band (ω ≈ −1.4ω_LO) the imaginary part vanishes because there exist no dark states with a smaller energy. If the vertex corrections are not taken into account, this lower threshold for the imaginary part is shifted toω =−ωLO since the spectrum of the dark stateshψ_vl^†_−pψ_clb^†_pitis now similar to the spectrum of a free electron-hole pair with no excitonic resonances. The peak atω ≈ −0.4ω_LO is due to the upper band edge of the valence band. It is conspicuous that the singularity at the lower edge of the excitonic band is not visible in Figure2.5. This is due to the fact that the differenceΦ^ex_k+q−Φ^ex_k which appears as a factor in the two sums of Equation2.103vanishes continuously in the limitq →0. Consequently, the contributions of the excitonic states with eigenen-ergiesω_x(q)near the lower band edge (q = 0) are suppressed. A similar explanation can be used in order to understand the missing singularity at ω = −ω_LO in Figure 2.6where the lower band edge of the free spectrum of the dark states hψ_vl−p^† ψclb^†_pi^tis located.

The effect of the self-energy corrections on the form of the susceptibility function χex(ω)can be studied in Figure2.7where its imaginary part is plotted with and with-out vertex corrections for two different temperatures. For comparison, the imaginary

−2 −1 0 1 ω/ω_LO

−0.2 −0.1 0.0 0.1

self−energy Sex(ω)/ωLO

real part imaginary part

Figure 2.5: The excitonic self-energyS_ex(ω)in the vicinity ofω_ex(0)at the tempera-turek_BT = 0.8ω_LO, calculated with vertex corrections

−2 −1 0 1

ω/ωLO

−0.2 −0.1 0.0 0.1

self−energy Sex(ω)/ωLO

real part imaginary part

Figure 2.6: The excitonic self-energyS_ex(ω)in the vicinity ofω_ex(0)at the tempera-turek_BT = 0.8ω_LO, calculated without vertex corrections

−1.5 −1 −0.5 0 ω/ωLO

−0.5

−1.0

−1.5

b)

−0.5

−1.0

−1.5

a)

Figure 2.7: The imaginary part of the excitonic susceptibility χ_ex(ω)at the tempera-tures a)k_BT = 0.4ω_LOand b)k_BT = 0.8ω_LO, calculated with (solid line) and without (dashed line) vertex corrections, the dotted line denotes the imaginary part of χex(ω) in the absence of the electron-phonon interaction

part of the susceptibility for a free exciton is also plotted where a small broadening has been added. If no vertex corrections are taken into account, the formation of a double peak structure at the shifted excitonic resonance can be observed with increasing tem-perature. This is due to the fact that the energy for the exciton with no center-of-mass momentum nearly coincides with the energy of the upper band edge of the valence band in the spectrum for the dark stateshψ^†_vl−pψ_clb^†_pit. If the vertex corrections are in-cluded, the exciton peak broadens with increasing temperature, but no splitting can be observed since the real and the imaginary part of the self-energy are only slowly vari-able in the vicinity ofω_ex(0). The resonance structure which appears below the exciton peak is due to the singularity of the self-energy at the upper edge of the excitonic band of the dark states (confer Figure2.5).

The explanation for the resonance structure below the excitonic energy is

corrob-orated, if the imaginary parts of the excitonic susceptibility functions for different electron-hole mass ratiosκ are compared (Figure2.8). In agreement with analytical results the upper band edge is shifted towards ω = −ω_LO + ω_ex(0) for decreasing κ. If the hole mass is infinite, the excitonic band has no dispersion and only a sharp resonance atω =−ω_LO +ω_ex(0)occurs.

−2 −1.5 −1 −0.5 0

ω/ω_LO

−1.5 −1 −0.5

Figure 2.8: The imaginary part of the excitonic susceptibilityχ_ex(ω)at the temper-aturek_BT = 1.2ω_LO, calculated with vertex corrections forκ = 0.15(dotted line), κ= 0.05(dashed line) andκ= 0(solid line)

In Reference [56] the imaginary part of the excitonic susceptibility function has been calculated for the same model and the same parameters which have been used here by solving the complete system of linear differential equations numerically with the help of the Lanczos algorithm. The resulting curves are in good agreement with the approximated ones from Figures2.7and2.8apart from small additional resonance structures below the upper edge of the excitonic band for the dark stateshψ^†_vl−pψ_clb^†_pit. The comparison shows that the qualitative features of the linear excitonic susceptibility are reproduced well within the framework of the diagonal approximation. In general, this is not true [21].

The phonon emission process which is responsible for the relaxation of highly

0.0 0.5 1.0 1.5 2.0 2.5 ω/ω_LO

δωk=1.75ωLO

δωk=1.50ωLO

δω_k=1.25ω_LO

δωk=1.00ωLO

δω_k=0.75ω_LO

Figure 2.9: The imaginary part of the susceptibilityχ_k(ω)at the temperaturek_BT = 0.4ω_LOfor different continuum energiesδω_k =ω_k−∆˜

excited electrons can be studied more closely by calculating the susceptibility

χ_k(ω) = − 1

ω+i0−ω_k−S_k(ω+i0) (2.104) for a continuum state with energyω_k (confer AppendixB). The self-energy function Sk is defined in the same way as the self-energy for the exciton, only the coefficients for the excitonic eigenvector, Φ^x_l, are replaced by the corresponding coefficients for the continuum state with energyω_k,Φ^k_l. In Figure2.9the imaginary part ofχ_k(ω)has been plotted for different energies ωk. If ωk is smaller than a characteristic threshold energy ω_t, the spectral function is peaked sharply at ω_k. This corresponds to a slow decay of the initial state. For energiesω_kwhich are in the vicinity ofω_tone observes a double peak structure which is due to the strong coupling between electron and phonon modes. Forω_k ω_tthe spectrum exhibits a broad Lorentzian peak at ω_k indicating a fast decay of the initial state. The existence of this threshold energy shows that the life-time of the continuum states is mainly determined by the influence of the phonon emission processes. A fast decay of excited continuum states can only be observed, if a

scattering into continuum states with a lower energy by phonon emission is consistent with the classical energy conservation. For this reason, one would expectω_tto be equal toω =ω_LO+ ∆ +U. The observed threshold energy, however, is located above this energy value. This discrepancy can be explained by the fact thatχ_k(ω)is the spectral function for the electron-hole pairs. Consequently, the frequencyωrefers to the pair energy and not to the energy of the conduction electrons alone.

The phenomenon of the shifted threshold energy can be explained analytically by calculating the susceptibilityχk(ω)for non-interacting electrons (U = 0andU˜ = 0) at zero temperature. Since the external field only excites electron-hole pairs with even parity the components of the eigenvector which describes the optically excited state with energyωk =E(k,0)are defined by

Φ^k_l = 1

√2{δ_l;k+δ_l;−k}. (2.105) If the self-energy function is evaluated at the resonanceω = ωk, the susceptibility is approximately given by

χ_k(ω)≈ − 1

ω+i0−ω_k−S_k(ω_k+i0). (2.106) In the thermodynamic limitS_k(ω_k+i0)satisfies the equation

Sk(ωk+i0) = g² 2π

Z π

−π

dq 1

ω_k+i0−ω_LO −_c;k+q+_v;k + g²

2π Z π

−π

dq 1

ωk+i0−ωLO −c;k+v;k+q

(2.107) and the imaginary part of the self-energy is only different from zero, if one of the two following conditions

4t_c+ ∆ +ω_LO ≥_c;k ≥∆ +ω_LO (2.108)

⇔4{t_c+t_v}+ ∆ +ω_LO{1 +κ} ≥ω_k ≥∆ +ω_LO{1 +κ} and

4t_v+ω_LO ≥ −_v;k ≥ω_LO (2.109)

⇔4{t_c+t_v}+ ∆ +ω_LO

1 +κ⁻¹ ≥ ω_k ≥∆ +ω_LO

1 +κ⁻¹

is met. If the parameters from Section 2.1 are used, only the first relation can be satisfied and the threshold energy is equal toωt =ωLO{1 +κ}in agreement with the experimental results of Reference [39].

Im Dokument Nonlinear Optical Effects in Pure and N-Doped Semiconductors (Seite 27-38)